Trapped modes in a waveguide with a thick obstacle

The problem of finding necessary and sufficient condi-tions for the existence of trapped modes in waveguides has been known since 1943. [10]. The problem is the following: consider an infinite strip M in xs211D2(or an infinite cylinder with the smooth boundary in xs211Dn). The spectrum of the(positive) Laplacian, with either Dirichlet or Neumann boundary conditions, acting on this strip is easily computable via the separation of variables; the spectrum is absolutely continuous and equals [v0,+∞). Here, v0 is the first threshold, i.e., eigenvalue of the cross-section of the cylinder (so v0 = 0 in the case of Neumann conditions). Let us now consider the domain S0025579300015606_inline1 (the waveguide) which is a smooth compact perturbation of M (for example, weinsert an obstacle inside M). The essential spectrum of the Laplacian acting on S0025579300015606_inline1 still equals [v0, +xs211D), but there may be additional eigenvalues, which are often called trapped modes; the number of these trapped modes can be quite large

Complete asymptotic expansion of the integrated density of states of multidimensional almost-periodic pseudo-differential operators

We obtain a complete asymptotic expansion of the integrated density of states of operators of the form H =(-\Delta)^w +B in R^d. Here w >0, and B belongs to a wide class of almost-periodic self-adjoint pseudo-differential operators of order less than 2w. In particular, we obtain such an expansion for magnetic Schr\"odinger operators with either smooth periodic or generic almost-periodic coefficients.Comment: 47 pages. arXiv admin note: text overlap with arXiv:1004.293

Asymptotic expansion of the integrated density of states of a two-dimensional periodic Schrodinger operator

We prove the complete asymptotic expansion of the integrated density of states of a two-dimensional Schrodinger operator with a smooth periodic potentialComment: 46 pages, 4 figure

Trapped modes in a waveguide with a long obstacle

Consider an infinite two-dimensional acoustic waveguide containing a long rectangular obstacle placed symmetrically with respect to the centreline. We search for trapped modes, i.e. modes of oscillation at particular frequencies which decay down the waveguide. We provide analytic estimates for trapped mode frequencies and prove that the number of trapped modes is asymptotically proportional to the length of the obstacle

Bethe-Sommerfeld conjecture for periodic operators with strong perturbations

We consider a periodic self-adjoint pseudo-differential operator $H=(-\Delta)^m+B$, $m>0$, in $\R^d$ which satisfies the following conditions: (i) the symbol of $B$ is smooth in \bx, and (ii) the perturbation $B$ has order less than $2m$. Under these assumptions, we prove that the spectrum of $H$ contains a half-line. This, in particular implies the Bethe-Sommerfeld Conjecture for the Schr\"odinger operator with a periodic magnetic potential in all dimensions.Comment: 61 page

Ballistic transport for Schrödinger operators with quasi-periodic potentials

We prove the existence of ballistic transport for a Schrödinger operator with a generic quasi-periodic potential in any dimension d > 1

Perturbative diagonalization for Maryland-type quasiperiodic operators with flat pieces

We consider quasiperiodic operators on Zd with unbounded monotone sampling functions ("Maryland-type"), which are not required to be strictly monotone and are allowed to have flat segments. Under several geometric conditions on the frequencies, lengths of the segments, and their positions, we show that these operators enjoy Anderson localization at large disorder

Complete asymptotic expansion of the spectral function of multidimensional almost-periodic Schrodinger operators

We prove the existence of a complete asymptotic expansion of the spectral function (the integral kernel of the spectral projection) of a Schrödinger operator H=−Δ+bH=−Δ+b acting in RdRd when the potential bb is real and either smooth periodic, or generic quasiperiodic (finite linear combination of exponentials), or belongs to a wide class of almost-periodic functions